Dr. Prabhakaran and co-authors here present a novel approach to spatial network analysis of fractures. This approach involves quantifying hierarchical clustering based on similarity of four statistics: fingerprint distance, D-measure, NetLSD, and portrait divergence. Hierarchical clusters are identified based on the similarity of areal sub-samples of a large fracture map (the example here being the Lilstock anticline, UK). The authors show maps of their results, noting apparent spatial autocorrelation in all analyses except for NetLSD.

The manuscript is mostly well written and appears to me to be mathematically sound. However, despite using several highly sophisticated statistics, the method relies on at least two rather qualitative and/or arbitrary choices: (i) the sampling window size and overlap amount, which would seem to exert an effect on the degree of the aforementioned spatial autocorrelation, and (ii) the selection of the number of clusters present, which should sensitively affect the resulting map patterns.

Moreover, the utility of the approach remains elusive, despite some time spent on it in the introduction and discussion. These four statistics are novel, but their meaning is fairly obscure. The main output of the technique is a map that highlights spatial variation in these statistics, which indeed could be useful. But the same types of patterns show up on maps of more inuitive statistics, like intensity (Figure 9, 14, 19). What does the technique tell you about the spatial arrangement of fractures, fracture connectivity, fracture drivers, etc., that exceeds the explanatory power of more conventional approaches?

Use of the term “clusters” (Line 96-99) is non-standard. Marrett et al. (2018), which the present paper cites, defines a cluster as a place where fracture spacing is smaller than it is elsewhere; as such, a uniformly spaced fracture pattern would lack clusters entirely. This definition is consistent with the framework described earlier in the present study (Line 26-7). Why the change in usage? I suggest using a different term for the output of the present approach that is not already widely used in pattern-quantification literature.

Line-by-line comments:

6: S/V agreement. Also 8, 10, 11, 289, 344.

21: ambiguous pronoun “they”

24: I suggest abandoning this acronym. It is only used once more in the paper (Line 396) and furthermore it does not stand for its definition.

27: See definition comment above; also, in this framework I believe that “irregularly spaced” is a synonym for “clustered” and “regularly spaced” is a synonym for “periodically spaced.”

42: None of these drawbacks is unique to 1D analysis.

57: I am not an expert on stationarity, but I don't think this is true. Can't an irregular pattern nevertheless be stationary, meaning have attributes that do not change with position, above a certain size-scale?

66: Also “called” nodes?

92: What do undirected and weighted mean?

Figure 2: The construction of this figure is very difficult to understand until we read about Algorithm 1 below.

106: HC undefined acronym until Line 250, and even then it should be made explicit.

124-30: I would add synkinematic cementation (Hooker and Katz, 2015, Am. J. Sci.) to this list.

125: “etc” inappropriate—if more studies have made the point and need to be listed, list them; else put “e.g.” and then your chosen, representative references.

Table 1. Are these Edges and Nodes in the primal or dual graph sense?

141: I don’t understand: how is curvature illustrated in these plots?

176: Greater regularity: can you justify this? A long, thin rectangular block would be highly “regular” and yet have a shape factor near zero.

179: “distribution of block-face regions”—you mean areas?

203: “w.r.t” seems needlessly curt; why not spell it out? Also 424.

258: “HC clusters” redundant

260: “decisions on the height”—see major comment above about the arbitrariness of this decision. You put a tremendous amount of effort into quantifying aspects of the spatial arrangement, and then seem to make an arbitrary decision as to how to bin your clusters.

297: “correctly” a loaded term without more explanation.

298: I would omit “clear” and let the reader judge. Also “clearly” in 315.

Figure 18 caption: acronyms not used in figure?

342: “unsupervised”—see major comment above about arbitrary choices. The user has to make the call about what qualifies as a cluster here; I don’t think this approach matches the spirit of unsupervised learning.

357: “striking” for “trending”

360: “background-variation”—such an interpretive term, especially considering the only external clustering control you have eliminated, to some degree, is faulting.

386: "we will tackle”—this is inappropriate. Anyone who wants to extend your work is entirely free to do so. I am sure you are not intentionally “marking your territory,” but we need to be especially cognizant of this kind of thing, especially in these days of pre-prints, self-archiving, predatory journals, and similar avenues whereby unscrupulous workers can “jump the gun” on an idea before proper peer review.

391: “inherent non-stationarity” see comment on Line 57

392: define “disassortativity”

397: “stationarity” for “stationariness”?

397: “have to be made based on hard data”—see major comment on arbitrarily choosing the number of clusters, above. Have you made your decisions based on hard data?

408: define “modularity”

411: What is graph partitioning, and is that a worthy goal?

418: “hierarchical” for “hierarchically”?

This paper is an interesting contribution to an important topic – the analysis spatial variation in fracture networks.  The paper introduces some methods, established in other areas but new to this area of earth science.  The work is based on some interesting field data, and is both well written and illustrated.  I recommend publication, but offer the following comments (see also comments on early sections in annotated pdf).

Section 1,2

The authors provide a concise, clear background to the treatment of fracture networks as graphs (Section 2).   The level of explanation is appropriate for the rest of the paper, but I highlight, in an annotated version of the manuscript, a few areas that the authors might want to clarify or omit.

Section 3

The introduction to the field area is similarly clear and concise.  There is no mention of a recent paper by Procter and Sanderson (2017) that discusses the spatial variability of fractures in the same geological units, just a few kilometres to the west. This paper not only uses graphs to represent the network, but also provides a statistical evaluation of the between layer and within layer variability of fracture intensity. 

Section 4: Methods

Section 4.2 discusses a range of graph measures used in the subsequent hierarchical clustering.  There is a lot of technical detail in the definition of these measures, which is difficult to follow but the sources are all clearly stated.  What would be most helpful to the reader would be an evaluation of what each measure is contributing in terms of the geometry and topology of the network. For example:

• The “fingerprint measure” clearly defines a block “shape”, both in terms of the number of sides and aspect ratio of the overall shape. Given that most block have 4-6 sides, an average aspect ratio would a little less than 2 and I would expect that this parameter would mainly be reflecting variation in aspect ratio. 
• The D-measure is mainly based on the clustering of the node distribution. Given that the divergence and alpha centrality seem to vary little, I would think this measure mainly reflects variation in the node intensity, which seems to be supported by Fig. 14.

It would be good to have a similar evaluation of the other measures. In particular, it is not clear to me how the variation in fracture orientation is captured by these measures.  Since the distribution of sets with differing orientation is a major feature of at least two of these regions (Passchier et al 2021), it is surprising that this aspect is omitted from description of the measures and appears to play little role in the clustering..

Section 5: Results

This provides a detailed analysis of three regions and presents the results of the mapping of spatial variability in terms of the HC of the measures used.  The results are presented through sets of five similar diagrams for each of the three regions.  Some of the material in these diagrams could be transferred to “supplementary material” (e.g. heat maps and dendrograms of the clustering).

It is probably worth comparing the characteristics of each region.  From Table 1, the number of fractures/branches/nodes, with similar ranges of fracture intensity (P21) in Figs 9, 14 and 19.  The average degrees are also very similar (~3.15) indicating a close approximation to a 3-regular graph (or mesh).  Apart from the “fingerprint” of block shape (Fig. 7e), there is no information on the distributions of the other measures used in the clustering.

 It seems to me, that Region 2 illustrates the strengths and weaknesses of the methodology, so starting with a complete analysis of this area would make sense.  The other two regions could then be treated more concisely, with emphasis on what they contribute to the study.

Section 6: Discussion

In the section on choice of graph metrics, the paragraph from lines 378-386 is key to this paper.  An unweighted graph cannot contain information on geometry, since a graph is invariant to change in shape and size.  Thus, geometry must be represented through the embedding of the graph in a (geographical) space or by including geometrical measures in the weights.  The former allows specification of orientation and length, and measure of the frequency and intensity of elements to unit area.  The way this paragraph is written implies that length and orientation could be incorporated but were not in the present study – this is “shooting one’s self in the foot”.

Conclusions and Abstract

 Both these sections are written in a vague way, expressing the aims and aspirations of the study, instead of focusing on the main findings.

David J Sanderson

Procter, A., Sanderson, D.J., 2017. Spatial and layer-controlled variability in fracture networks. Journal of Structural Geology doi.org/10.1016/j.jsg.2017.07.008.